Finite difference methods for 2D and 3D wave equations
hplgit.github.io › num-methods-for-PDEs › docMany applications involve variable coefficients, and the general wave equation in d dimensions is in this case written as. (105) ϱ ∂ 2 u ∂ t 2 = ∇ ⋅ ( q ∇ u) + f for x ∈ Ω ⊂ R d, t ∈ ( 0, T], which in, e.g., 2D becomes. (106) ϱ ( x, y) ∂ 2 u ∂ t 2 = ∂ ∂ x ( q ( x, y) ∂ u ∂ x) + ∂ ∂ y ( q ( x, y) ∂ u ∂ y) + f ( x, y, t).
Finite difference methods for 2D and 3D wave equations
hplgit.github.io › fdm-book › docWe shall now describe in detail various Python implementations for solving a standard 2D, linear wave equation with constant wave velocity and \(u=0\) on the boundary. The wave equation is to be solved in the space-time domain \(\Omega\times (0,T]\), where \(\Omega = (0,L_x)\times (0,L_y)\) is a rectangular spatial domain. More precisely, the complete initial-boundary value problem is defined by